Black Hole Entropy, Special Geometry and Strings

Abstract

We review work done over the last years on the macroscopic and microscopic entropy of supersymmetric black holes in fourdimensional N=2 supergravity and in N=2 compactifications of string theory and M-theory. Particular emphasis is put on the crucial role of higher curvature terms and of modifications of the area law in obtaining agreement between the macroscopic entropy, which is a geometric property of black hole solutions and the microscopic entropy, which is computed by state counting in Calabi-Yau compactifications of string or M-theory. We also discuss invariance properties of the entropy under stringy T-duality and S-duality transformations in N=2,4 compactifications in presence of higher curvature terms. In order to make the paper self-contained we review the laws of black hole mechanics in higher derivative gravity, the definition of entropy as a surface charge, the superconformal off-shell description of N=2 supergravity, special geometry, and N=2 compactifications of heterotic and type II string theory and of M-theory.